#7888956

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4311 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,8)(3,6,4,7)(11,13)(16,18)(19,21)(23,24)(25,26)(27,34)(28,32)(29,33)(30,35)(31,38)(36,37)(39,40);(1,2)(3,6)(4,7)(5,8)(9,10,15,12)(11,18,13,16)(14,17,22,20)(19,24,21,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,40)(34,35);(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,26)(27,34)(28,32)(29,33)(30,35)(31,38)(36,37)(39,40);(1,3,5,4)(2,6,8,7)(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24)(25,28,30,33)(26,32,35,29)(27,36,38,40)(31,39,34,37);(9,13,15,11)(10,16,12,18)(14,19,22,21)(17,24,20,23)(25,29,30,32)(26,33,35,28)(27,37,38,39)(31,40,34,36);(1,4,5,3)(2,7,8,6)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,27)(26,34)(28,36)(29,39)(30,38)(31,35)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,e^2=i,f^2=ghi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7796152);